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== '''ARTICLE START''' ==
= '''ARTICLE START''' =
'''Telicity''' is a property of both [[equal temperament]]s and [[comma]]s and how they relate to each other. An edo is p-2 '''telic''' when it tempers a comma in 2.p subgroup for a prime p, and that comma is smaller than half an edostep.
'''Telicity''' is a property of both [[equal temperament]]s and [[comma]]s and how they relate to each other. An edo is p-2 '''telic''' when it tempers a comma in 2.p subgroup for a prime p, and that comma is smaller than half an edostep.


Commas and equal temperaments that demonstrate this property are referred to as as being '''telic'''. When a given EDO is telic in a given multiprime relationship by more than one means, it can be said to be '''multitelic'''.
Commas and equal temperaments that demonstrate this property are referred to as as being '''telic'''. When a given EDO is telic in a given multiprime relationship by more than one means, it can be said to be '''multitelic'''.


== Telicity and Continued fractions ==
== Telicity and Continued fractions ==
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<math>k\cdot d_{enominator}\left(C\right)\left|d_{enominator}\left(C\right)\log_{m}\left(n\right)-n_{umerator}\left(C\right)\right|<\frac{1}{2}</math>
<math>k\cdot d_{enominator}\left(C\right)\left|d_{enominator}\left(C\right)\log_{m}\left(n\right)-n_{umerator}\left(C\right)\right|<\frac{1}{2}</math>


Multitelicity is not the same as having many telicities. For example, 12edo is 3-2 telic and 5-2 telic, but only multitelic in the 3-2.
Multitelicity is not the same as having multiple telicities in different primes. For example, 12edo is 3-2 telic and 5-2 telic, but only multitelic in the 3-2. To be more precise, 12edo is 5-2 telic, 2-strong 3-2 telic.
 
=== Pseudotelicity ===
If the p2-p1-telic circle in an edN reaches other (normally higher) primes [p3] with less than 25% error, then the edN is p3;p2-p1-pseudotelic. To cite some examples:
 
In 53edo, since the primes 5 and 7 connect to the telic chain of fifths with less than 25% error, then it is 7-5/3-2 pseudotelic. The prime 5 itself can be stacked four times, but not the 7, so it is more precisely 7-5⁴;3-2 pseudotelic.
 
In 159edo, since the prime 11, stacked thrice, connects to the telic chain of fifths with less than 25% error, so it is 11³-3-2 pseudotelic.


=== Examples ===
=== Examples ===
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* 1/[[1edo|1]], 2/1, 3/[[2edo|2]], 8/[[5edo|5]], 19/[[12edo|12]], 65/[[41edo|41]], 84/[[53edo|53]], 485/[[306edo|306]], 1054/[[665edo|665]], 24727/[[15601edo|15601]], 50508/[[31867edo|31867]], 125743/[[79335edo|79335]], 176251/[[111202edo|111202]]...
* 1/[[1edo|1]], 2/1, 3/[[2edo|2]], 8/[[5edo|5]], 19/[[12edo|12]], 65/[[41edo|41]], 84/[[53edo|53]], 485/[[306edo|306]], 1054/[[665edo|665]], 24727/[[15601edo|15601]], 50508/[[31867edo|31867]], 125743/[[79335edo|79335]], 176251/[[111202edo|111202]]...


The commas that arise from these edos are the following, with the corresponding :
The commas that arise from these edos are the following, with the corresponding:


* 1edo: [[4/3]] [<nowiki/>[[Bixby|bixby, degenerate case]]]
* 1edo: [[4/3]] [<nowiki/>[[Bixby|bixby, trivial]]]
* 2edo: [[9/8]] [<nowiki/>[[Very low accuracy temperaments#Antitonic|antitonic, degenerate case]]]
* 2edo: [[9/8]] [<nowiki/>[[Very low accuracy temperaments#Antitonic|antitonic]]]
* 5edo: [[256/243]] [<nowiki/>[[blackwood]]]
* 5edo: [[256/243]] [<nowiki/>[[blackwood]]]
* 12edo: [[Pythagorean comma|531441/524288]] [<nowiki/>[[compton]]]
* 12edo: [[Pythagorean comma|531441/524288]] [<nowiki/>[[compton]]]
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Of those, 12, 53, 665 are multitelic, because they have a k-strength value greater than one; being 2, 3, and 11 respectively, which means that [[24edo|24]], [[106edo|106]], [[159edo|159]], [[1330edo|1330]], [[1995edo|1995]], [[2660edo|2660]], [[3325edo|3325]], [[3990edo|3990]], [[4655edo|4655]], [[5320edo|5320]], [[5985edo|5985]], [[6650edo|6650]], and [[7315edo|7315]] are also 3-2 telic.
Of those, 12, 53, 665 are multitelic, because they have a k-strength value greater than one; being 2, 3, and 11 respectively, which means that [[24edo|24]], [[106edo|106]], [[159edo|159]], [[1330edo|1330]], [[1995edo|1995]], [[2660edo|2660]], [[3325edo|3325]], [[3990edo|3990]], [[4655edo|4655]], [[5320edo|5320]], [[5985edo|5985]], [[6650edo|6650]], and [[7315edo|7315]] are also 3-2 telic.
A naïve way find if an edo is p-2 telic, multiply the relative error of that prime by the edo. If the error is less than 50%, it is telic.


== Applications ==
== Applications ==
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Non-telic edos with convergent fifths, or semiconvergent fifths (which are never telic), are also incredibly good and offer comparably great intervals, specially if the edo is big enough. Edos with semiconvergent fifths include [[7edo|7]], [[17edo|17]], [[29edo|29]], [[94edo|94]], [[147edo|147]], [[200edo|200]], [[253edo|253]], [[359edo|359]], [[971edo|971]]...
Non-telic edos with convergent fifths, or semiconvergent fifths (which are never telic), are also incredibly good and offer comparably great intervals, specially if the edo is big enough. Edos with semiconvergent fifths include [[7edo|7]], [[17edo|17]], [[29edo|29]], [[94edo|94]], [[147edo|147]], [[200edo|200]], [[253edo|253]], [[359edo|359]], [[971edo|971]]...
Adding telic edos together results in semiconvergents when the smaller one is added to the bigger one or multiples thereof, like 94 = 53 + 41; 17 = 12 + 5. However, adding the smaller convergent more than once, or subtracting it, also results in edos with good fifths and possibly great properties. For this reason, the term "Continued Fraction Deviant" [CFD] may be used to extend the notion of semiconvergents, into adding telic edos together more generally.


=== MOS ===
=== MOS ===
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|+
|+
|3-2 telic
|3-2 telic
| rowspan="5" |1
|[[2edo|2]]
|2
|'''[[5edo|5]]'''
|'''5'''
|'''[[12edo|12]]''' [<nowiki/>[[24edo|'''24''']]]
|'''12''' [24]
|'''[[53edo|53]]''' [<nowiki/>[[159edo|'''159''']]]  
|'''53''' [159]  
3-k
3-k
|306 [''612'']
|[[306edo|306]] ''[<nowiki/>[[612edo|'''612''']]]''
|'''665''' [7315]  
|'''[[665edo|665]]''' [<nowiki/>[[7315edo|'''7315''']]]  
11-k
11-k
|15601 ''[78005]''
|[[15601edo|'''15601''']] ''[<nowiki/>[[78005edo|'''78005''']]]''
|31867
|[[31867edo|31867]]
|79335
|[[79335edo|79335]]
|190537  
|[[190537edo|190537]]
28-k
28-k
|-
|-
|5-2 telic
|5-2 telic
|3 [12]
|[[3edo|3]] [<nowiki/>[[12edo|'''12''']]]
4-k
4-k
|28
|[[28edo|28]]
|59
|[[59edo|59]] [<nowiki/>[[118edo|'''118''']]]
|146
|[[146edo|146]]
2-k
2-k
|'''643'''
|'''[[643edo|643]]'''
3-k
3-k
|4004
|[[4004edo|4004]]
|8651
|[[8651edo|8651]]
|12655
|[[12655edo|12655]]
|21306
|[[21306edo|21306]]
2-k
2-k
|97879
|[[97879edo|97879]]
9-k
9-k
|-
|-
|7-2 telic
|7-2 telic
|5 [10, ''15'']
|'''[[5edo|5]]''' ['''[[10edo|10]]''']
|26 [52, ''130'']
2-k
|109 [218]
|'''[[26edo|26]]''' [''[[130edo|'''130''']]'']
|571
2-k
|[[109edo|109]]
2-k
|[[571edo|571]]
2-k
2-k
|2964
|[[2964edo|2964]]
15-k
15-k
|91313
|[[91313edo|91313]]
2-k
2-k
|453601
|[[453601edo|453601]]
4-k
4-k
|
|
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|-
|-
|11-2 telic
|11-2 telic
|2
|[[2edo|2]]
3-k
3-k
|13 [26]
|[[13edo|13]] [<nowiki/>[[26edo|'''26''']]]
|37
2-k
|[[37edo|37]]
13-k
13-k
|986
|[[986edo|986]]
|1935
|[[1935edo|1935]]
|4856
|[[4856edo|4856]]
|16503
|[[16503edo|16503]]
12-k
12-k
|
|
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|-
|-
|13-2 telic
|13-2 telic
|1
|[[3edo|3]]
|3
|[[7edo|'''7''']]
|7
|'''[[10edo|10]]''' [<nowiki/>[[50edo|'''50''']], [[80edo|'''80''']], ''[[130edo|'''130''']], [[270edo|'''270''']]'']
|10 [50, 80, ''130, 270'']&nbsp;
11-k
11-k
|227 [908]
|[[227edo|227]] [<nowiki/>[[908edo|908]]]
11-k
11-k
|5458
|[[5458edo|5458]]
4-k
4-k
|54353
|[[54353edo|54353]]
74-k
74-k
|
|
|
|
|
|
|
|}
{| class="wikitable"
|+Table of 3-convergency
|CFD-2
|CFD-1
|Convergent
|CFD+1
|CFD+2
|CFD+3
|-
|
|
|1
|
|
|
|-
|
|(1)
|2
|3
|4
|(5)
|-
|(1)
|3
|'''5'''
|'''7'''
|9
|11
|-
|6
|'''8'''
|'''''10'''''
|(12)
|(14)
|16
|-
|(2)
|(7)
|'''12'''
|17
|'''22'''
|'''27'''
|-
|14
|'''19'''
|'''''24'''''
|29
|'''34'''
|39
|-
|'''26'''
|'''31'''
|''36''
|(41)
|'''46'''
|51
|-
|(17)
|(29)
|'''41'''
|(53)
|'''65'''
|77
|-
|
|(12)
|'''53'''
|'''94'''
|'''135'''
|176
|-
|(24)
|'''(65)'''
|''106''
|147
|188
|229
|-
|(77)
|'''118'''
|'''''159'''''
|200
|241
|'''282'''
|-
|'''130'''
|'''171'''
|''212''
|253
|294
|335
|-
|'''183'''
|'''224'''
|''265''
|(306)
|347
|388
|-
|(200)
|(253)
|306
|359
|412
|465
|-
|506
|559
|''612''
|(665)
|718
|771
|-
|(53)
|(359)
|665
|971
|1277
|1583
|-
|718
|1024
|''1330''
|'''1636'''
|1942
|2248
|-
|1383
|1689
|''1995''
|2301
|2607
|3003
|}
|}
Bolded edos are notable. Edos in brackets are notable telic edo multiples, due to high consistency or accuracy of their intervals. If italicized, the edo is a multiple of the telic edo but beyond its k-strength range.
1edo and 0edo excluded. Bolded telic edos are notable. Edos in brackets are notable telic edo multiples, due to high consistency or accuracy of their intervals. If italicized, the edo is a multiple of the telic edo but beyond its k-strength range.
 
Of these, the only edos ''(that I know so far)'' that are telic in many primes are 3edo (13-5-2 telic), 5edo (7-3-2 telic), 12edo (5-3-2 telic), and 26edo (11-7-2 telic).


WIP
WIP
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